The Power of a Statistical Test

Date: 02/11/99 at 08:10:53
From: John Kohut III
Subject: Power of a statistical test
Dear Dr. Math,
Is there a formula to determine the power of a statistical test which
relates alpha and beta error?
For example, does the sample group of numbers 12, 14, 15, 16, 20
belong to the population of numbers with a mean of 17 with standard
deviation of 2? What would be the power of the z-test performed on
this sample using alpha = 95% and beta = 80%? Thank you.
John Kohut III

Date: 02/11/99 at 10:50:02
From: Doctor Anthony
Subject: Re: Power of a statistical test
Below is an example of how the power of a test is calculated.
A population is known to have a variance of 9. Investigate whether the
population mean is equal to 10. The sample size is 49 and the
probability of a type I error is taken to be alpha = .1. If the true
value of the population mean is 8, calculate the power of the test.
If we reject the null hypothesis when it is true we make a type 1
error, and its probability is denoted by alpha.
The power of the test is the probability of rejecting the hypothesis
when it is false, and it is denoted by 1 - beta.
With a .1 significance level, the probability of rejecting the null
hypothesis will be 0.05 in each tail, so we use z = +/- 1.645.
If the null hypothesis is true, the mean is 10. We want to find an
acceptance region where:
P(reject Ho | Ho is true) = P(reject Ho | mean = 10) = .1
Using the null hypothesis of mean = 10, the acceptance region is given
by
xm - mean xm - 10
+/-1.645 = -------------------------- = -------
std. dev/sqrt(sample size) 3/7
So the acceptance region would be
xm = 10 +/- (3/7)(1.645)
which is
9.295 < xm < 10.705
The probability of rejecting the null hypothesis when the null is
false (i.e. mean = 8) is given by the z-value
9.295 - 8 1.295
z = --------- = ----- = 3.02
3/7 3/7
which has a probability of
A(z) = 0.9987
So, in this situation, the power of the test is 0.9987.
- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/